Subgroups — #18011521

Order 4736.

398 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14558585	C₂	1	2	yes	maximal, complemented
2	#14558565	C₂	1	2	yes	maximal, complemented
3	#14558564	C₂	1	2	yes	maximal, complemented
4	#14558560	C₂	1	2	yes	maximal, complemented
5	#14558555	C₂	1	1	yes	maximal, characteristic, complemented
6	#14558554	C₂	1	1	yes	maximal, characteristic, complemented
7	#14558549	C₂	1	1	yes	maximal, characteristic, complemented
8	#14558536	C₂	1	1	yes	maximal, characteristic, complemented
9	D₃₇⋊(C₄⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄⋊#64438	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#10596964	C₂	1	1	yes	maximal, characteristic
12	C₂×#10596957	C₂²	1	2	yes	—
13	C₂×D₃₇⋊C₈	C₂²	1	2	yes	—
14	C₂²×#64438	C₂²	1	2	yes	—
15	#11924603	C₂²	1	2	yes	—
16	#11924603	C₂²	1	2	yes	—
17	C₄⋊#64438	C₂²	1	1	yes	characteristic
18	C₄⋊#64438	C₂²	1	1	yes	characteristic
19	C₄⋊#64438	C₂²	1	2	yes	—
20	C₄⋊#64438	C₂²	1	1	yes	characteristic
21	C₄⋊#64438	C₂²	1	1	yes	characteristic
22	#11924593	C₂²	1	2	yes	—
23	C₂²⋊#64438	C₂²	1	2	yes	—
24	C₂²⋊#64438	C₂²	1	2	yes	—
25	#11924591	C₂²	1	2	yes	—
26	#11924591	C₂²	1	2	yes	—
27	C₄×#64438	C₂²	1	2	yes	—
28	C₂×#10596964	C₄	1	1	yes	characteristic
29	C₄×Dic₇₄	C₄	1	1	yes	characteristic
30	#11924501	C₂²	1	1	yes	characteristic
31	D₄×Dic₃₇	C₂²	1	1	yes	characteristic
32	C₄×C₂²⋊D₃₇	C₄	1	1	yes	characteristic
33	C₄×C₂²⋊D₃₇	C₄	1	1	yes	characteristic
34	#11924496	C₂²	1	1	yes	characteristic
35	#11924496	C₂²	1	1	yes	characteristic
36	#11924494	C₄	1	1	yes	characteristic
37	#11924492	C₄	1	1	yes	characteristic
38	#11924492	C₄	1	1	yes	characteristic
39	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
40	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
41	C₄²×D₃₇	C₂²	1	1	yes	characteristic
42	C₄×D₄×C₃₇	C₄	1	1	yes	characteristic
43	C₂×D₄×C₃₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₂²⋊D₃₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₂²⋊D₃₇	C₂×C₄	1	1	yes	characteristic
48	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
49	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
50	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
51	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
52	#10596964	—	2	4	no	—
53	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
56	C₂×Dic₇₄	C₂×C₄	1	1	yes	characteristic
57	#10596957	—	4	2	no	—
58	C₂×#64438	—	4	2	no	—
59	C₂×#64438	C₂³	1	2	yes	—
60	C₂×#64438	C₂³	1	2	yes	—
61	C₂×#64438	C₂³	1	2	yes	—
62	C₂×#64438	C₂³	1	2	yes	—
63	D₃₇⋊C₈	—	4	2	no	—
64	C₄⋊C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
65	C₂²⋊C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
66	C₂²⋊C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
67	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
68	C₂²⋊Dic₃₇	C₂×C₄	1	1	yes	characteristic
69	C₂²⋊Dic₃₇	C₂×C₄	1	1	yes	characteristic
70	#10596938	C₂×C₄	1	1	yes	characteristic
71	#10596938	C₂×C₄	1	1	yes	characteristic
72	C₄⋊Dic₃₇	C₂×C₄	1	1	yes	characteristic
73	#10596936	C₂×C₄	1	1	yes	characteristic
74	#10596936	C₂×C₄	1	1	yes	characteristic
75	#10596936	C₂×C₄	1	1	yes	characteristic
76	#10596936	C₂×C₄	1	1	yes	characteristic
77	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
78	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
79	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
80	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
81	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
82	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
83	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
84	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
85	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
86	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
87	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
88	C₂²×D₃₇	C₂²×C₄	1	1	yes	characteristic
89	D₄×C₃₇	—	2	2	no	—
90	C₂×C₄×C₃₇	—	2	2	no	—
91	C₂×C₄×C₃₇	—	2	2	no	—
92	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
97	C₂²⋊D₃₇	—	2	2	no	—
98	C₂²⋊D₃₇	M₁₆	1	4	yes	complemented
99	C₂×Dic₃₇	—	2	2	no	—
100	C₂×Dic₃₇	—	2	2	no	—
101	C₂×Dic₃₇	—	2	1	no	—
102	C₂×Dic₃₇	—	2	1	no	—
103	C₂×Dic₃₇	—	2	1	no	—
104	C₂×Dic₃₇	—	2	1	no	—
105	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
106	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
107	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
108	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
109	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
110	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
111	C₂×Dic₃₇	C₂⁴	1	1	yes	characteristic
112	C₄×D₃₇	—	2	2	no	—
113	C₄×D₃₇	—	2	2	no	—
114	C₄×D₃₇	—	2	1	no	—
115	C₄×D₃₇	—	2	1	no	—
116	Dic₇₄	—	2	2	no	—
117	#64438	—	2	2	no	—
118	#64438	—	2	2	no	—
119	#64438	—	2	2	no	—
120	#64438	—	2	2	no	—
121	C₂²×C₃₇	—	2	1	no	—
122	C₂²×C₃₇	—	2	1	no	—
123	C₂²×C₃₇	—	2	1	no	—
124	C₂²×C₃₇	—	2	1	no	—
125	C₂²×C₃₇	—	2	1	no	—
126	C₂²×C₃₇	C₂×M₁₆	1	2	yes	—
127	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
128	C₂×D₃₇	—	2	1	no	—
129	C₂×D₃₇	—	2	1	no	—
130	C₂×D₃₇	C₂×M₁₆	1	2	yes	—
131	C₄×C₃₇	—	2	1	no	—
132	C₄×C₃₇	—	2	1	no	—
133	C₄×C₃₇	—	2	1	no	—
134	C₄×C₃₇	—	2	1	no	—
135	C₄×C₃₇	#131	1	2	yes	—
136	Dic₃₇	—	2	1	no	—
137	Dic₃₇	—	2	1	no	—
138	Dic₃₇	—	2	1	no	—
139	Dic₃₇	—	2	1	no	—
140	Dic₃₇	—	2	1	no	—
141	Dic₃₇	C₂×M₁₆	1	2	yes	—
142	Dic₃₇	H(5,𝔽₂)	1	1	yes	characteristic
143	Dic₃₇	2₋¹⁺⁴	1	1	yes	characteristic
144	#2990	—	37	1	no	maximal, Sylow
145	C₂×C₃₇	—	2	2	no	—
146	C₂×C₃₇	—	4	1	no	—
147	C₂×C₃₇	#520	1	1	yes	characteristic
148	C₂×C₃₇	C₂²×M₁₆	1	1	yes	characteristic
149	C₂×C₃₇	C₂×#131	1	1	yes	characteristic
150	D₃₇	—	2	2	no	—
151	C₄×#41	—	37	1	no	—
152	#446	—	37	2	no	—
153	#435	—	37	2	no	—
154	#435	—	37	2	no	—
155	C₈×D₄	—	37	2	no	—
156	#432	—	37	1	no	—
157	#432	—	37	1	no	—
158	#431	—	37	1	no	—
159	#423	—	37	1	no	—
160	C₂×C₄⋊C₈	—	37	1	no	—
161	C₂×C₄⋊C₈	—	37	1	no	—
162	C₃₇	#2990	1	1	yes	characteristic, complemented, Sylow
163	C₂×#41	—	37	1	no	—
164	C₂×M₁₆	—	37	2	no	—
165	C₂²×C₈	—	37	2	no	—
166	C₂²×C₈	—	37	2	no	—
167	C₄×Q₈	—	37	1	no	—
168	C₄×Q₈	—	37	1	no	—
169	C₄×D₄	—	37	1	no	—
170	C₄×D₄	—	37	1	no	—
171	C₄×D₄	—	37	1	no	—
172	C₄×D₄	—	37	1	no	—
173	C₄×D₄	—	37	1	no	—
174	C₄×D₄	#3866	1	1	yes	characteristic
175	#117	—	37	1	no	—
176	#117	—	37	1	no	—
177	#117	—	37	1	no	—
178	C₂×C₄²	—	37	1	no	—
179	C₂×C₄²	—	37	1	no	—
180	C₂×C₄²	—	37	1	no	—
181	C₄⋊C₈	—	37	1	no	—
182	C₄⋊C₈	—	37	1	no	—
183	C₄⋊C₈	—	37	1	no	—
184	C₄⋊C₈	—	37	2	no	—
185	C₄⋊C₈	—	37	1	no	—
186	C₄⋊C₈	—	37	2	no	—
187	C₄⋊C₈	—	37	2	no	—
188	C₂²⋊C₈	—	37	2	no	—
189	C₂²⋊C₈	—	37	2	no	—
190	C₂²⋊C₈	—	37	2	no	—
191	#97	—	37	2	no	—
192	#97	—	37	2	no	—
193	C₄×C₈	—	37	2	no	—
194	#41	—	74	4	no	—
195	C₂×Q₈	—	37	1	no	—
196	C₂×D₄	—	37	1	no	—
197	C₂×D₄	—	37	1	no	—
198	C₂×D₄	C₂×#3866	1	1	yes	characteristic
199	C₂²×C₄	—	37	1	no	—
200	C₂²×C₄	—	37	1	no	—
201	C₂²×C₄	—	37	1	no	—
202	C₂²×C₄	—	37	1	no	—
203	C₂²×C₄	—	37	1	no	—
204	C₂²×C₄	—	37	1	no	—
205	C₂²×C₄	—	37	1	no	—
206	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
207	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
208	M₁₆	—	148	2	no	—
209	C₂×C₈	—	37	2	no	—
210	C₂×C₈	—	37	2	no	—
211	C₂×C₈	—	37	2	no	—
212	C₂×C₈	—	37	2	no	—
213	C₂×C₈	—	148	2	no	—
214	C₂×C₈	—	148	2	no	—
215	C₄⋊C₄	—	37	1	no	—
216	C₄⋊C₄	—	37	1	no	—
217	C₄⋊C₄	—	37	1	no	—
218	C₄⋊C₄	—	37	1	no	—
219	C₄⋊C₄	—	37	1	no	—
220	C₄⋊C₄	C₂×#3866	1	1	yes	characteristic
221	C₂²⋊C₄	—	37	1	no	—
222	C₂²⋊C₄	—	37	1	no	—
223	C₂²⋊C₄	—	37	1	no	—
224	C₂²⋊C₄	—	37	1	no	—
225	C₂²⋊C₄	C₂×#3866	1	1	yes	characteristic
226	C₂²⋊C₄	C₂×#3866	1	1	yes	characteristic
227	C₄²	—	37	1	no	—
228	C₄²	—	37	1	no	—
229	C₄²	—	37	1	no	—
230	C₄²	—	37	1	no	—
231	C₄²	—	37	1	no	—
232	C₄²	—	37	1	no	—
233	C₄²	—	37	1	no	—
234	C₄²	—	37	1	no	—
235	C₄²	—	37	1	no	—
236	C₄²	C₂×#3866	1	1	yes	characteristic
237	C₂³	—	37	1	no	—
238	C₂³	C₂²×#3866	1	1	yes	characteristic
239	C₂³	C₂²×#3866	1	1	yes	characteristic
240	Q₈	—	74	2	no	—
241	D₄	—	2	2	no	—
242	D₄	—	37	4	no	—
243	D₄	—	74	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	37	1	no	—
247	C₂×C₄	—	37	1	no	—
248	C₂×C₄	—	37	1	no	—
249	C₂×C₄	—	37	1	no	—
250	C₂×C₄	—	37	1	no	—
251	C₂×C₄	—	37	1	no	—
252	C₂×C₄	—	37	1	no	—
253	C₂×C₄	—	74	1	no	—
254	C₂×C₄	—	74	1	no	—
255	C₂×C₄	—	74	2	no	—
256	C₂×C₄	—	74	1	no	—
257	C₂×C₄	—	74	1	no	—
258	C₂×C₄	—	74	2	no	—
259	C₂×C₄	—	74	1	no	—
260	C₂×C₄	—	74	1	no	—
261	C₂×C₄	—	74	2	no	—
262	C₂×C₄	—	74	2	no	—
263	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
264	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
265	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
266	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
267	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
268	C₈	—	74	2	no	—
269	C₈	—	74	2	no	—
270	C₈	—	74	2	no	—
271	C₈	—	74	2	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	37	2	no	—
278	C₂²	—	74	1	no	—
279	C₂²	—	74	1	no	—
280	C₂²	D₃₇⋊M₁₆	1	2	yes	—
281	C₂²	C₂³×#3866	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	37	1	no	—
287	C₄	—	37	1	no	—
288	C₄	—	37	2	no	—
289	C₄	—	74	1	no	—
290	C₄	—	74	1	no	—
291	C₄	—	74	1	no	—
292	C₄	—	74	1	no	—
293	C₄	—	74	1	no	—
294	C₄	#11924623	1	2	yes	—
295	C₂	—	2	2	no	—
296	C₂	—	4	1	no	—
297	C₂	—	74	2	no	—
298	C₂	#14558602	1	1	yes	characteristic
299	C₂	C₂×D₃₇⋊M₁₆	1	1	yes	characteristic
300	C₂	C₂×#11924623	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#3866
commutator, socle	C₂²×C₃₇	C₂³×C₄
Fitting	C₄×D₄×C₃₇	C₄
prime radical	C₂×#10596964	C₄
cosocle	C₂×Dic₃₇	C₂⁴
hypercenter	C₄×D₄	#3866
hypocenter	C₃₇	#2990
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #14558585	C₂	semidirect
2. #14558565	C₂	semidirect
3. #14558564	C₂	semidirect
4. #14558560	C₂	semidirect
5. #14558555	C₂	semidirect
6. #14558554	C₂	semidirect
7. #14558549	C₂	semidirect
8. #14558536	C₂	semidirect
9. D₃₇⋊(C₄⋊C₈)	C₂	semidirect
10. C₂×C₄⋊#64438	C₂	semidirect
98. C₂²⋊D₃₇	M₁₆	semidirect
162. C₃₇	#2990	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#14558585	2
#14558565	2
#14558564	2
#14558560	2
#14558555	2
#14558554	2
#14558549	2
#14558536	2
D₃₇⋊(C₄⋊C₈)	2
C₂×C₄⋊#64438	2
C₄×#10596964	2
#2990	37